Fast Log-Likelihood Ratio (LLR) Computation for Decoding High-Order and High-Dimensional Modulation Schemes

ABSTRACT

A method receives the symbol transmitted over a channel, selects, from a constellation of codewords, a first codeword neighboring the received symbol and a set of second codewords neighboring the first codeword, and determines a relative likelihood of each second codeword being the transmitted symbol with respect to a likelihood of the first codeword being the transmitted symbol. Next, the method determines an approximation of a log-likelihood ratio (LLR) of each data bit in the received symbol as a log of a ratio of a sum of the relative likelihoods of at least some of the second codewords having the same value of the data bit to a sum of the relative likelihoods of at least some of the second codewords having different value of the data bit and decodes the received symbol using the LLR of each data bit.

FIELD OF THE INVENTION

The present invention relates generally to digital communications, andmore specifically to computing log-likelihood ratio (LLRs) for decodingmodulated symbols.

BACKGROUND OF THE INVENTION

In a digital communication system, a transmitter typically encodestraffic data based on a forward-error correction (FEC) coding scheme toobtain code bits and further maps the code bits to modulation symbolsbased on a modulation scheme. The transmitter then processes themodulation symbols to generate a modulated signal and transmits thissignal via a communication channel. The communication channel distortsthe transmitted signal with a channel response and further degrades thesignal with noise and interference.

A receiver receives the transmitted signal and processes the receivedsignal to obtain symbols, which can be distorted and noisy versions ofthe modulation symbols sent by the transmitter. The receiver can thencompute log-likelihood ratio (LLRs) for the code bits based on thereceived symbols. The LLRs are indicative of the confidence in zero(‘0’) or one (‘1’) being sent for each code bit. For a given code bit, apositive LLR value can indicate more confidence in ‘0’ being sent forthe code bit, a negative LLR value can indicate more confidence in ‘1’being sent for the code bit, and an LLR value of zero can indicate equallikelihood of ‘0’ or ‘1’ being sent for the code bit. With FEC decoder,the receiver can then decode the LLRs to obtain decoded data, which isan estimate of the traffic data sent by the transmitter.

In a soft-in soft-out (SISO) decoder, “soft” refers to the fact that theincoming and/or outgoing data can take on values other than 0 or 1, inorder to indicate reliability. The soft output is the LLR for value ofthe bit, is used as the soft input to an outer decoder. The computationfor the LLRs can be complex, leading to high power consumption. However,accurate LLRs can increase the decoding performance. There is thereforea need in the art for techniques to efficiently and accurately computeLLRs for code bits.

The computational difficulty for LLRs calculation is even more apparentfor optical communication systems using block-coded high-dimensionalmodulation formats. For example, any digital modulation scheme uses afinite number of distinct symbols to represent digital data. Forexample, conventional dual-polarization binary phase-shift keying(DP-BPSK) transmits two bits on the four dimensions of the opticalcarrier. Two dimensions are modulated independently, and only twodimensions are utilized. However, the optical coherent communicationsystems are naturally suited for modulation with four-dimensional (4D)signal constellations.

Four-dimensional modulation formats can achieve substantial gainscompared with conventional modulation formats such as dual-polarizationquaternary phase-shift keying (DP-QPSK) and 16-ary quadrature-amplitudemodulation (DP-16QAM). Polarization-switched QPSK (PS-QPSK) andset-partitioned 128-ary QAM (SP-128QAM) are known to be practical 4Dconstellations, and they can achieve 1.76 dB and 2.43 dB gains inasymptotic power efficiency, respectively. The achievable gain can befurther improved by using higher dimensional modulation formats. Forexample, 24D extended Golay code achieves 6.00 dB gain by producing theblock of 24 bits including 12 data bits and 12 parity bits selected toincrease distance between possible symbols, which makes the LLRscalculation computationally complex.

Accordingly, there is a need to reduce computational complexity fordetermining LLRs for modulated symbols transmitted over a channel.

SUMMARY OF THE INVENTION

It is an object of some embodiments of an invention to provide a systemand a method for decoding modulated symbols transmitted over a channel,such as a wireless channel or an optical channel. In some embodiments ofthe invention, a modulated symbol includes data bits and parity bitsused for encoding the symbol.

Some embodiments of the invention provide a system and a method fordetermining LLRs for modulated symbols in a computationally efficientmanner. For example, some embodiments trade off accuracy of the LLRcalculation against computational complexity by considering a subset ofthe most likely nearest neighbor codewords for each bit of transmittedsymbol.

Specifically, some embodiments are based on recognition that the LLRs ofall codewords can be calculated to determine the most likely codeword.However, the LLR can also be approximated by using the relativelikelihood of the codewords with respect to another codeword. Inaddition, some embodiments are based on a realization that byconsidering only nearest neighbor codewords an almost exact LLR can beapproximated while significantly reducing the number of calculations.For example, in one embodiment of the invention for 24D extended Golaycode, the number of calculations for determining the LLRs of modulatedsymbols is reduced from 4096 to 759 by considering the likelihoods ofonly nearest neighbors.

In some embodiments of the invention, the LLR approximation is based onbelief propagation over factor graph for block-coded high-dimensionalmodulations, which can be expressed by a parity-check matrix. The outputof LLR values from the belief propagation can be further refined by anonlinear filter, including artificial neural networks and Volterrafilter. In order to reduce the computational complexity of nonlinearfilters, some minor edges over nonlinear filter are pruned. Stochasticback-propagation provides more accurate LLR approximation byreinforcement learning offline. In addition, one embodiment takessoft-decision feedback from the FEC decoder to refine LLRs as abit-interleaved coded modulation with iterative demodulation. Theoffline learning is carried out for BICM-ID to provide more accurate LLRapproximations over graph. Another embodiment of the invention usesnonbinary FEC coding. The method of the LLR approximation is generalizedfor any Galois field size, where multiple LLR values are treated as oneLLR vector message.

In yet another embodiment, the method of the invention providesefficient high-dimensional modulation to maximize the mutual informationof the approximated LLRs. The method projects N-dimensional hyper cubesonto M-dimensional subspace to achieve shaping gain for M-dimensionalmodulation formats. One embodiment uses an exponential mapping matrixfor Grassmannian manifold.

Accordingly, one embodiment of the invention discloses a method fordecoding a symbol transmitted over a channel, wherein the symbol isencoded and modulated to include data bits and parity bits, includingreceiving the symbol transmitted over a channel, wherein the receivedsymbol includes the transmitted symbol modified with noise of thechannel; selecting, from a constellation of codewords, a first codewordneighboring the received symbol and a set of second codewordsneighboring the first codeword; determining a relative likelihood ofeach second codeword being the transmitted symbol with respect to alikelihood of the first codeword being the transmitted symbol;determining an approximation of a log-likelihood ratio (LLR) of eachdata bit in the received symbol as a log of a ratio of a sum of therelative likelihoods of at least some of the second codewords having thesame value of the data bit to a sum of the relative likelihoods of atleast some of the second codewords having different value of the databit; and decoding the received symbol using the LLR of each data bit.The steps of method are performed using a processor of a decoder.

Another embodiment discloses a receiver decoding a symbol transmittedover a channel, wherein the symbol is encoded and modulated to includedata bits and parity bits, including: a demodulator connected to anantenna to receive the symbol transmitted over a channel, wherein thereceived symbol includes the transmitted symbol modified with noise ofthe channel; a memory to store a constellation of codewords; and adecoder connected to a processor to select, from the constellation ofcodewords, a first codeword neighboring the received symbol and a set ofsecond codewords neighboring the first codeword, to determine a relativelikelihood of each second codeword being the transmitted symbol withrespect to a likelihood of the first codeword being the transmittedsymbol, to determine an approximation of a log-likelihood ratio (LLR) ofeach data bit in the received symbol as a log of a ratio of a sum of therelative likelihoods of at least some of the second codewords having thesame value of the data bit to a sum of the relative likelihoods of atleast some of the second codewords having different value of the databit, and to decode the received symbol using the LLR of each data bit.

BRIEF DESCRIPTION OF THE DRAWING

FIG. 1 is a block diagram of a design of a transmitter 100 and areceiver150 in a digital communication system employed by someembodiments of the invention;

FIG. 2 is a block diagram of the encoder and the symbol mapper at thetransmitter in FIG. 1 according to one embodiment of the invention;

FIG. 3 is a schematic of an exemplar signal constellation used by oneembodiment of the invention;

FIG. 4 is a block diagram of a method for decoding a symbol transmittedover a channel according to some embodiments of the invention;

FIG. 5 is a schematic of the constellation of codewords including thereceived symbol mapped to the constellation according to someembodiments of the invention;

FIG. 6A is a block diagram of a method for determining an approximationof the log-likelihood ratio (LLR) for each data bit of the receivedsymbol according to one embodiment of the invention;

FIG. 6B is a schematic of possible groupings of symbols with three databits constellation used by one embodiment of the invention;

FIG. 7 is a block diagram of a system and/or a method for modulating anoptical signal according to some embodiments of the invention;

FIG. 8 is a schematic of an exemplar mapping of the basis vector to acarrier in a time domain according to some embodiments of the invention;and

FIG. 9 is a block diagram of a method for approximating LLR calculationassuming according to one embodiment of the invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

FIG. 1 shows a block diagram of a design of a transmitter 100 and areceiver150 in a digital communication system. At transmitter 100, anencoder 120 receives a block of data from a data source 112, encodes thedata block based on an FEC coding scheme, and provides code bits. A datablock can also be referred to as a transport block, a packet, or aframe. An encoder 120 can perform rate matching and delete or repeatsome or all of the code bits to obtain a desired number of code bits forthe data block. The encoder 120 can also perform channel interleavingand reorder the code bits based on an interleaving scheme. A symbolmapper 130 maps the code bits to modulation symbols based on amodulation scheme, which may be QPSK, QAM, etc. A modulator (MOD) 132can perform processing for code-division multiplexing (CDM),frequency-division multiplexing (FDM), orthogonal frequency-divisionmultiplexing (OFDM), single-carrier FDM (SC-FDM), or single-carrier(SC). The modulator 132 then processes, e.g., converts to analog,amplifies, filters, and frequency up-converts, the resultant outputsymbols and generates modulated symbols, which are transmitted via anantenna 134 for wireless communication systems. For opticalcommunication systems, electro-optical devices such as laser are used totransmit corresponding symbols over fiber.

At receiver 150, an antenna 152 receives the modulated signal fromtransmitter 100 and provides a received signal. A demodulator (DEMOD)154 processes, e.g., filters, amplifies, frequency down-converts, anddigitizes, the received signals to obtain discrete samples. Demodulator154 can further process the samples (e.g., for CDM, FDM, OFDM, SC-FDM,etc.) to obtain received symbols.

A signal and noise estimator 162 can estimate signal and noisecharacteristics and/or the wireless/optical channel response based onthe received symbols. An LLR computation unit 160 computes LLRs for codebits based on the received symbols and the signal, noise and/or channelestimates. A decoder 170 decodes the LLRs in a manner complementary tothe encoding performed by transmitter 100 and provides decoded data. Ingeneral, the processing by demodulator 154, LLR computation unit 160,and decoder 170 at receiver 150 is complementary to the processing bymodulator 132, symbol mapper 130, and encoder 120 at transmitter 100.

Controllers/processors 140 and 180 direct the operation of variousprocessing units at transmitter 100 and receiver 150, respectively. Thememories 142 and 182 store data, e.g., a constellation of codewords, andprogram codes for transmitter 100 and receiver 150. In general, encoder120 can implement any FEC coding scheme, such as a turbo code, aconvolutional code, a low-density parity-check (LDPC) code, a cyclicredundancy check (CRC) code, a block code, etc., or a combinationthereof. The encoder 120 can generate and append a CRC value to a datablock, which can be used by the receiver 150 to determine whether thedata block decoded correctly or in error. Turbo code, convolutionalcode, and LDPC code are different FEC codes that allow the receiver 150to correct errors caused by impairments in the wireless or opticalchannels.

FIG. 2 shows a block diagram of the encoder 120 and the symbol mapper130 at the transmitter 100 in FIG. 1 according to one embodiment of theinvention. In this embodiment, the encoder 120 implements a turbo code,which is also referred to as a parallel concatenated convolutional code.Within encoder 120, a code interleaver 222 receives a block of data bits(denoted as {d}) and interleaves the data bits in accordance with a codeinterleaving scheme. A first constituent encoder 220 a encodes the databits based on a first constituent code and provides first parity bits(denoted as {z}). A second constituent encoder 220 b encodes theinterleaved data bits from code interleaver 222 based on a secondconstituent code and provides second parity bits (denoted as {z′}).According to other embodiments, the encoder 120 may implement a lowdensity parity check (LDPC) code, a staircase code, a BCH code, a polarcode, or any other channel coding scheme.

For example, the constituent encoders 220 a and 220 b can implement twogenerator polynomials, e.g., g₀(D)=1+D²+D³ and g₁(D)=1+D+D³ used inWideband Code Division Multiple Access (W-CDMA), where D denotes a delayoperator. A multiplexer (Mux) 224 receives the data bits and the paritybits from constituent encoders 220 a and 220 b, multiplexes the data andparity bits, and provides code bits. Multiplexer 224 may cycle throughits three inputs and provide one bit at a time to its output, or {d₁,z₁, z′₁, d₂, z₂, z′₂, . . . }. A rate matching unit 226 receives thecode bits from multiplexer 224 and may delete some of the code bitsand/or repeat some or all of the code bits to obtain a desired number ofcode bits for the data block. Although not shown in FIG. 2, encoder 120may also perform channel interleaving on the code bits from ratematching unit 226.

In some embodiments, within symbol mapper 130, a demultiplexer (Demux)230 receives the code bits from encoder 120 and demultiplexes the codebits into an in-phase (I) stream {i} and a quadrature (Q) stream {q}.Demultiplexer 230 can provide the first code bit to the I stream, thenthe next code bit to the Q stream, then the next code bit to the Istream, etc. A QAM/QPSK look-up table 232 receives the I and Q streams,forms sets of B bits, and maps each set of B bits to a modulation symbolbased on a selected modulation scheme, where B=2 for QPSK, B=4 for16-QAM, etc. Symbol mapper 130 provides modulation symbols {x} for thedata block.

FIG. 3 shows an example signal constellation for 16-QAM, which is usedby some embodiments of the invention. This signal constellation includes16 signal points corresponding to 16 possible modulation symbols for16-QAM. Each modulation symbol is a complex value of the formx_(i)+jx_(q), where x_(i) is the real component, x_(q) is the imaginarycomponent, and j denotes imaginary unit. The real component x_(i) canhave a value of −3α, −α, α or 3α, and the imaginary component x_(q) canalso have a value of −3α, −α, α or 3α, where a is typically 1/sqrt(10).

For 16-QAM, the code bits in the I and Q streams from demultiplexer 230can be grouped into sets of four bits, with each set being denoted as{i₁ q₁ i₂ q₂ }, where bits i₁ and i₂ are from the I stream and bits q₁and q₂ are from the Q stream. The 16 modulation symbols in the signalconstellation are associated with 16 possible 4-bit values for {i₁ q₁ i₂q₂}. FIG. 3 shows an example mapping of each possible 4-bit value to aspecific modulation symbol. In this mapping, the real component x_(i) ofa modulation symbol is determined by the two in-phase bits i₁ and i₂,and the imaginary component x_(q) is determined by the two quadraturebits q₁ and q₂. In particular, bit i₁ determines the sign of the realcomponent x_(i), with x_(i)>0 for i₁=0, and x_(i)<0 for i₁=1. Bit i₂determines the magnitude of the real component x_(i), with |x_(i)|=a fori₂=0, and |x_(j)|=3a for i₂=1. Bit i₁ may thus be considered as a signbit for x_(i), and bit i₂ may be considered as a magnitude bit forx_(i). Similarly, bit q₁ determines the sign of the imaginary componentx_(q), and bit q₂ determines the magnitude of the imaginary componentx_(q). The mapping is independent for the real and imaginary components.For each component, 2-bit values of ‘11’, ‘10’, ‘00’ and ‘01’ are mappedto −3α, −α, α, and 3α, respectively, based on pulse amplitude modulation(PAM). Two 4-PAM modulation symbols may thus be generated separatelybased on (i₁ i₂) and (q₁ q₂) and then quadrature combined to obtain a16-QAM modulation symbol.

Higher-order modulation formats such as 64-QAM and 256-QAM receive morebits to generate modulated symbols. Increasing the modulation orderusually requires more complicated demodulation to produce LLRs for thedecoder because the number of constellation points is of an exponentialorder with respect of the number of bits. In some embodiments, themodulator uses higher-dimensional modulation formats to improveresilience against channel noise. For such high-dimensional modulation,even more complex computation for demodulation is required. The methodof the invention provides computationally efficient LLR calculations forlow-power demodulation in particular for high-order and high-dimensionalmodulation formats.

LLR Calculation

FIG. 4 shows a block diagram of a method for decoding a symboltransmitted over a channel according to some embodiments of theinvention. The symbol is encoded and modulated to include data bits andparity bits, and can be transmitted over a wireless, wired, orfiber-optic channel. The method decodes the transmitted symbol bydetermining an approximation of LLR of each data bit as a logarithm of aratio of a sum of the relative likelihoods of at least some of thesecond codewords having the same value of the data bit to a sum of therelative likelihoods of at least some of the second codewords havingdifferent value of the data bit. Steps of the method can be using aprocessor 401 of a decoder.

The method receives 410 the symbol 415 transmitted over a channel. Thereceived symbol includes the transmitted symbol modified with noise ofthe channel. For example, the received symbol includes data bitsmodified with noise with known received values and unknown transmittedvalues selected from a predetermined constellation of codewords 460.Examples of such constellations include the constellation 310 of FIG. 3.Next, the method selects, from a constellation of codewords 460, a firstcodeword neighboring the received symbol and a set of second codewords425 neighboring the first codeword.

FIG. 5 shows a schematic of the constellation of codewords 460 includingthe received symbol 415 mapped to the constellation. Some embodiments ofthe invention determine the first codeword 520 as a codeword closest tothe received symbol. For example, one embodiment determines the firstcodeword 520 using soft-in hard-out (SIHO) decoding of the symbol. Anexample of the SIHO decoding is minimum Euclidean distance decoding.

Some embodiments are based on recognition that the LLR can beapproximated using relative likelihood of the most likely nearestneighbor symbol 530 and the first codeword 520. However, such approachsuffers from an approximation error. To that end, some embodiments ofthe invention determine a plurality of the codewords having a distanceto the first codeword less or equal a threshold to form the set ofsecond codewords. In such a manner, the set of second codewords includenot only the most likely nearest neighbor symbol 530, but also othersymbols 510 with non-negligible likelihoods if the given bit is in errorin the first codeword, and a plurality of symbols 515 other than thefirst codeword with non-negligible likelihood if the given bit iscorrect in the first codeword.

For example, some embodiments of the invention measure the distance tothe first codeword as a Hamming distance or as a Lie distance. Theembodiment is advantageous, because it is possible to design mappingssuch that there is a simple correspondence between Hamming or Liedistance and Euclidean distance. This enables selection of symbols whichare neighboring in Euclidean distance without having to directlycalculate Euclidean distance, which can be highly complex with largeconstellations in multiple dimensions. One embodiment of the inventiondetermines the threshold as a minimal distance between the firstcodeword and any other codeword in the constellation. As a result, allcodewords selected to the set of second codewords have the samedistance, e.g., Hamming distance, to the first codeword. This embodimentis useful for decoding blocks of modulated symbols that providesufficient number of second codewords with minimal distance to the firstcodeword. For example, for decoding a signal modulated with a24-dimensional (24D) format over a 4D optical carrier, there can be 759second codewords with minimum Hamming weight with respect to the firstcodeword.

Next, the method determines 430 a relative likelihood 435 of each secondcodeword being the transmitted symbol with respect to a likelihood ofthe first codeword being the transmitted symbol and determines 440 anapproximation of a log-likelihood ratio (LLR) 445 of each data bit inthe received symbol as a logarithm of a ratio of a sum of the relativelikelihoods of at least some of the second codewords having the samevalue of the data bit to a sum of the relative likelihoods of at leastsome of the second codewords having different value of the data bit.Next, the method decodes 450 the received symbol using the LLR 445 ofeach data bit to produce the decoded symbol 455, wherein steps of methodare performed using a processor of a decoder.

More specifically, some embodiments determine the LLR value L of a givenbit b_(j) within a received symbol x according to:

${{L\left( {b_{j} = {0x}} \right)} = {\log\left( \frac{\sum\limits_{x^{\prime} \in P_{j}}{S\left( {x = {{x^{\prime}b_{j}} = 0}} \right)}}{\sum\limits_{x^{\prime} \in P_{j}}{S\left( {x = {{x^{\prime}b_{j}} = 1}} \right)}} \right)}},$

wherein symbol likelihoods S are calculated for each possible symbol x′within a set of symbols P_(j). Two subsets are formed from the set ofsymbols—a set for which b_(j)=0; and a set for which b_(j)=1.

In the presence of Gaussian noise over channel, the symbol likelihood Sfor a specific possible symbol x′ is given by:

${{S\left( {x = x^{\prime}} \right)} = {\exp\left( \frac{- {{x - x^{\prime}}}^{2}}{\sigma^{2}} \right)}},$

wherein σ² is the noise variance.

In some situations, the exact calculation of soft-information can beprohibitively complex for high-dimensional modulation formats based onblock codes. By utilizing the output of a hard decision decoder, someembodiments of the invention approximate the soft-information byconsidering only the nearest neighbors of the hard-decision codeword.Additionally, one embodiment considers only orthogonal dimensions inwhich the second codewords differ from the hard-decision first codeword.In this embodiment, the number of elements which comprise thesoft-information is reduced.

The number of second codewords comprising the nearest neighbors forwhich the soft information is considered can be varied, with thecomplexity of the soft-output decoder becoming higher as this number isincreased. In some embodiments, subsequent processing such as sorting orminimum searching is used to reduce the complexity of calculating thesoft-information output. The calculation of LLR is then performed basedon the most significant subset of symbol likelihoods, therefore reducingcomputational complexity.

FIG. 6A shows a block diagram of a method for determining anapproximation of the LLR for each data bit of the received symbolaccording to one embodiment of the invention. The method forms, for eachdata bit of the received symbol, a group of the set of second codewords,such that there is one group for each data bit of the received symbol.The group formed such that a value of the data bits in each codeword inthe group on a position of the corresponding data bit of the receivedsymbol equals a value of the corresponding data bit of the receivedsymbol.

FIG. 6B shows the possible groupings of symbols 660 with three databits. Two groupings are made with the first bit being 0 (670) and 1(675) respectively. Two further groupings are made with the second bitbeing 0 (680) and 1 (685) respectively. Two final groupings are madewith the third bit being 0 (690) and 1 (695) respectively.

The method sorts 620 the second codewords in each group based on theircorresponding relative likelihood. By sorting the possible symbols bylikelihood, we are able to ignore symbols which have negligiblelikelihood from the bit LLR calculation. This is important, as mostsymbols have negligible likelihood.

The method sums 630, for each group, a predetermined number of the mostlikely second codewords and adds 640 one to a summation of a grouphaving the same data bit as in the first codeword. The predeterminednumber is chosen in the design phase as a tradeoff between computationalcomplexity and performance—choosing more symbols from the list willimprove the accuracy of the LLR approximation, while increasingcomputational complexity.

The method determines 650 a logarithm of a ratio for a pair of groupscorresponding to the same data bit (0 or 1) to produce 655 anapproximation of the LLR for each data bit of the received symbol. TheLLR is now approximated by a smaller number of subsets of symbollikelihoods, which give non-negligible likelihood values.

High-Dimensional Modulation for Coherent Optical Communications

Some embodiments of the invention determine the LLRs for opticalcommunication using block-coded high-dimensional modulation formats,such as modulation with 24-dimensional (24D) signal constellations. Inthose embodiments the large number of signaling dimensions causes thenumber of possible symbols in the modulation format to become extremelylarge (2¹²). In turn, this means that the number of symbol likelihoodsthat must be calculated is also extremely large (2¹²). For each subsetper bit in the LLR calculation, a large summation must also becalculated (2¹¹). Therefore, it is advantageous to reduce the complexityof this operation while maintaining accuracy at the given signal tonoise ratio.

FIG. 7 shows a block diagram of a system and/or a method for modulatingan optical signal according to embodiments of the invention. The systemincludes a transmitter 700 connected to a receiver 700 by an opticalfiber channel 750.

At the transmitter, data from a source 701 is outer encoded 710. Theouter encoder adds FEC parity redundancy 715. Then, a block encoder isapplied to an output of the outer encoder to produce encoded data 725.The block encoding is designed to increase the Hamming or Lie distancesbetween constellation points that represent the data. A mapper 730increases the Euclidian distances between constellation points toproduce mapped data 735. Then, the code, in the form of the mapped datacan be modulated 740 to a modulated signal that is transmitted throughthe optical channel 750. The transmission can use densewavelength-division multiplexing (WDM), multi-mode spatial multiplexing,multi-core spatial multiplexing, sub-carrier signaling, single-carriersignaling, and combination thereof

At the receiver, the steps of the transmitter are performed in a reverseorder, wherein the modulated signal is demoduled, demappedg,block-decoding, and FEC decoded to recover the data. Specifically,front-end processing 710 and channel equalization 720 are applied to thereceived optical modulated signal. A block decision 730 is made to feedthe soft-decision information to outer decoding 740 to recover the datafor a data sink 702.

To transmit the optical signal modulated with a 24-dimensional (24D)format over a 4D optical carrier, we map a 24D orthogonal signal vectorto a 4D optical carrier. To do so, we consider in-phase,quadrature-phase, polarization, and time as orthogonal dimensions. Insome embodiments, 24D orthogonal signal vector is mapped to additionalorthogonal dimensions such as spatial modes and frequencies.

FIG. 8 shows an example mapping of 24D basis vector (D₁, . . . , D₂₄) tothe 4D carrier in a time domain, where E_(XI) is the in-phase componentof the optical carrier on the horizontal polarization, E_(XQ) is thequadrature component of the optical carrier on the horizontalpolarization, E_(YI) is the in-phase component on the verticalpolarization, and E_(Yq) is the quadrature component on the verticalpolarization.

In a hypercube constellation, i.e., a constellation where each dimensionhas a value ±1 that is independent of all other dimensions and everydimension is bit-labeled independently; the squared Euclidean distancebetween constellation points is linearly proportional to the Hammingdistance. Therefore, we use a code designed to increase the Hammingdistance and the Euclidean distance between constellation points. Takingadvantage of this effect, we use the extended Golay code to determine asubset of the 24D hypercube. Then, the subset determines ourconstellation.

The extended Golay code encodes 12 bits of information into a 24-bitword with a minimum Hamming distance of 8. While this code has been usedwith an appropriate decoding matrix to correct for errors in wirelesscommunication and memories, we take maximum-likelihood (ML) decisions in24D to maintain soft information for an FEC decoder.

Although conventional ML decisions for a 12 bit word in 24D are usuallyhighly complex, we use a low-complexity demodulation of such formats,e.g., a multiplier free procedure based on correlation metriccalculation. It is also possible to use a lattice decoding or spheredecoding to reduce the complexity, which enables a practicalimplementation of the invention for short block sizes and real-timeprocessing.

In Golay-coded 24D modulation, the 2¹² points that correspond to validextended Golay codewords are our constellation points, from a possible2²⁴ points on the 24D hypercube. The minimum squared Euclidean distanceincreases by a factor of 8 compared with the 24D hypercube, which hasidentical performance to that of DP-QPSK, while the mean energy per bitis doubled. Therefore, asymptotic power efficiency is increased by 6 dBcompared with the 24D hypercube. The transmitter and receiver can besimilar those used with DP-QPSK modulation because the constellation isa subset of the hypercube.

FIG. 9 shows a block diagram of a method for approximating LLRcalculation assuming Golay-coded 24-D modulation according to oneembodiment of the invention. A 24-D input vector 910 is used tocalculate an initial soft-input hard-output decision 920. The bits b_(k)from this decision are then used to invert 925 the elements E_(k) of theinput vector 910, to make a vector of new elements M_(k) according tothe rule M_(k)=E_(k)*(−1)^(b) _(k). Following 925, a pre-determined listof minimum Hamming weight codewords 927 is used. The list of codewords927 is denoted as L, which has 759 codewords of weight 8 in the case ofthe 24-D extended Golay code. The weight vector of each codeword in L isthen calculated 930 by selecting the 8 elements M_(k) from themanipulated vector, for which the corresponding bit l_(k) in thecodeword from L is 1. The weight vectors are then summed 935 over their8 constituent elements, to create a single, scalar value for each of the759 codewords. The list of codewords L is then formed 940 into 12bipartite lists L_(j1) and L_(j0). These lists are defined by thecodewords on list L for which bit position j is 1, and 0 respectively.L_(j1) therefore has 506 elements, while L_(j0) has 253 elements,therefore each pair of lists contains a weight corresponding to all Lcodewords. Each list is then sorted by value 950, and arranged fromminimum to maximum. A fixed number of elements is then selected fromeach list, starting with the minimum 955. Each element is thenmultiplied by a constant 960, such that the log-symbol-likelihood,relative to the SIHO decision is produced. The log-sum-exponentialfunction is then used on both lists, with an additional element of ‘1’being appended to the lists drawn from L_(j0), to producelog-likelihoods W_(j0) and W_(j1). Each pair of log-likelihoods is thenused in combination with the SIHO decisions to produce alog-likelihood-ratio 970, R_(j), according toR_(j)=(W_(j0)−W_(j1))*(−1)^(b) _(k).

Another embodiment uses a single parity-check code to increase theHamming distance for 8D hypercube lattice modulations. The 7-bit dataare encoded by a block encoder to generate 8-bit coded word. Each bit ismodulated by BPSK per dimension, and then 8-dimensional BPSK mapped tothe 4D optical carrier. The decoder procedure is same as the previousembodiment. The benefit of the 8D modulation is lower complex in boththe encoder and the decoder.

Another embodiment uses near-perfect block codes, which offers themaximum possible Hamming distance over the hypercube lattice for atarget data rate and dimensions. Near-perfect block codes include linearand nonlinear codes such as Nordstrom-Robinson code, or combinations ofnear-perfect codes. Using hypercube lattice, the increase of the Hammingdistance can lead to the increase of Euclidean distance.Higher-dimensional lattice modulation can achieve better decoding forsignals subject to linear and nonlinear noise.

An alternative embodiment maps the constellation to the 4D opticalcarrier using a densest hypersphere lattice. The block code is designedby greedy sphere cutting to sequentially select the closes points overhigh-dimensional lattice point.

LLR Approximation over Graph and Projection Shaping

In another embodiment, the block decision can be done usingsoft-information belief propagation over a graphical representation(factor graph) of the block codes. High-dimensional modulation based onblock codes can be represented by a factor graph, in which parity-checkequation is described by check nodes connecting to associated bitvariable nodes. Using regular belief propagation based on sum-productalgorithm over the factor graph, the LLR can be approximated. However,the factor graph is not well optimized for belief propagation becausethe parity-check matrix is usually not sparse and there are many cycles.The method of the invention uses the belief messages over one or twoiterations over the factor graph as an initial estimate for the furtherprocessing to be more accurate. In one embodiment, the initial estimateof the LLRs including second nearest neighbors is modified by means ofartificial neural network for nonlinear filtering. The method uses alarge number of simulated received symbols for varying noise variance tolearn the neural networks in order to produce accurate LLR output.

For example, the reinforcement learning based on stochasticback-propagation or linear discriminant analysis can be used toapproximate LLR. The learning phase can be done offline. In addition,some edges in the neural networks are pruned so that the LLR calculationcan be low complex. In another embodiment, another nonlinear filteringbased on Volterra filter can be used to approximate the LLR. In yetanother embodiment, the LLR approximation can be performed for nonbinaryFEC coding, by treating multiple LLR values as a vectorized beliefmessage.

The method can be used for bit-interleaved coded modulation (BICM) withiterative demodulation (ID), which employs iterative demodulation givensoft-decision feedback from the FEC decoder. By feeding back thesoft-decision information, the demodulator can produce more accurateLLRs. For this embodiment, the reinforcement learning is carried outoffline by simulating noisy channels as well as various reliabilitylevels of soft-decision information. The neural networks now usesoft-decision information from the decoder in addition to the initialbelief messages as input data. Given the true LLR values, the neuralnetworks are stochastically learned by the back-propagation toapproximate the LLR values over the network propagation.

In yet another embodiment, high-order and high-dimensional modulationformats are designed to maximize the mutual information of approximatedLLR. The mutual information is calculated as follows:

=1−

log₂(1+exp(−L))

where L is the output of approximated LLRs, and E denotes theexpectation. The method of the invention uses a parametric projection ofblock-coded hyper-cube onto a subspace. For example, the projectionbased on Grassmannian exponential mapping is used to generateM-dimensional modulation formats from N-bit codeword. Any M-dimensionalsubspace of N-dimensional signals can be represented by M(N−M)parameters in theory. Such manifold can be expressed by matrixexponential function as follows:

$I_{M \times N} \times {\exp \left( \begin{bmatrix}0_{M} & \Theta \\{- \Theta} & 0_{N - M}\end{bmatrix} \right)}$

where I_(M×N) denotes the rectangular identity matrix of size M-by-N,0_(M) denotes all-zero matrix of size M-by-M, and the theta matrix is areal-valued matrix of size M-by-(N−M). For example, 2-bit onto1-dimensional modulation can be obtained by one parameter, and theGrassmannian projection of Gray-coded 2D square can be any arbitraryshaping of 4-PAM. The four signal points of the parameterized 4-PAMbecome −cos(theta)−sin(theta), −cos(theta)+sin(theta),cos(theta)−sin(theta), and cos(theta)+sin(theta). When theta isarctan(1/5), the 4-PAM signal can be regular 4-PAM (i.e., −3α, −α, α,and 3α) to generate 16-QAM, where two 4-PAM symbols are mapped ontoin-phase and quadrature components independently. By adjusting the thetavalue as a function of noise variance, the method of invention canimprove the mutual information by achieving shaping gain. The optimaltheta can be obtained offline by evaluating the mutual information ofthe approximated LLR. The optimized theta is also different whennonbinary LLR approximation is used. The Grassmannian projection ofblock-coded hyper cube can be used to generate even higher-ordermodulations and also higher-dimensional modulations. In one embodiment,the projection matrix is Stiefel manifold.

Although the invention has been described by way of examples ofpreferred embodiments, it is to be understood that various otheradaptations and modifications can be made within the spirit and scope ofthe invention. Therefore, it is the object of the appended claims tocover all such variations and modifications as come within the truespirit and scope of the invention.

We claim:
 1. A method for decoding a symbol transmitted over a channel,wherein the symbol is encoded and modulated to include data bits andparity bits, comprising: receiving the symbol transmitted over achannel, wherein the received symbol includes the transmitted symbolmodified with noise of the channel; selecting, from a constellation ofcodewords, a first codeword neighboring the received symbol and a set ofsecond codewords neighboring the first codeword; determining a relativelikelihood of each second codeword being the transmitted symbol withrespect to a likelihood of the first codeword being the transmittedsymbol; determining an approximation of a log-likelihood ratio (LLR) ofeach data bit in the received symbol as a log of a ratio of a sum of therelative likelihoods of at least some of the second codewords having thesame value of the data bit to a sum of the relative likelihoods of atleast some of the second codewords having different value of the databit; and decoding the received symbol using the LLR of each data bit,wherein steps of method are performed using a processor of a decoder. 2.The method of claim 1, wherein the symbols include data bits modifiedwith noise with known received values and unknown transmitted values. 3.The method of claim 1, further comprising: determining the firstcodeword using soft-in hard-out (SIHO) decoding of the symbol; anddetermining a plurality of the codewords having a distance to the firstcodeword less or equal a threshold to form the set of second codewords.4. The method of claim 3, wherein the distance is Hamming distance. 5.The method of claim 3, wherein the distance is Lie distance.
 6. Themethod of claim 3, further comprising: determining the threshold as aminimal distance between the first codeword and any other codewords inthe constellation.
 7. The method of claim 1, further comprising:forming, for each data bit of the received symbol, a group of the set ofsecond codewords, such that there is one group for each data bit of thereceived symbol, wherein a value of the data bits in each codeword inthe group on a position of the corresponding data bit of the receivedsymbol equals a value of the corresponding data bit of the receivedsymbol; sorting the second codewords in each group based on theircorresponding relative likelihood; summing, for each group, apredetermined number of the most likely second codewords; adding one toa summation of a group having the same data bit as in the firstcodeword; and determining a log of a ratio for a pair of groupscorresponding to the same data bit to produce an approximation of theLLR for each data bit of the received symbol.
 8. The method of claim 1,wherein the symbol transmitted, further comprising encoding a block codeto increase Euclidean distance; mapping in multi-dimensional hypercubes; and projecting onto multi-dimensional subspaces for shaping,wherein the projection matrix is a parametric manifold whose parametersare optimized offline so that the mutual information of the approximatedLLRs are maximized.
 9. The method of claim 1, wherein approximating theLLRs uses belief propagation over factor graph, and further refining theLLRs by nonlinear filters, wherein the nonlinear filter is optimizedoffline by learning algorithms.
 10. The method of claim 1, wherein LLRsare vectorized for nonbinary coding and decoding.
 11. The method ofclaim 8, wherein the nonlinear filter is learned by received symbols andsoft-decision feedback from the decoder to refine LLR values foriterative demodulation.
 12. A receiver decoding a symbol transmittedover a channel, wherein the symbol is encoded and modulated to includedata bits and parity bits, comprising: a demodulator connected to anantenna to receive the symbol transmitted over a channel, wherein thereceived symbol includes the transmitted symbol modified with noise ofthe channel; a memory to store a constellation of codewords; and adecoder connected to a processor to select, from the constellation ofcodewords, a first codeword neighboring the received symbol and a set ofsecond codewords neighboring the first codeword, to determine a relativelikelihood of each second codeword being the transmitted symbol withrespect to a likelihood of the first codeword being the transmittedsymbol, to determine an approximation of a log-likelihood ratio (LLR) ofeach data bit in the received symbol as a log of a ratio of a sum of therelative likelihoods of at least some of the second codewords having thesame value of the data bit to a sum of the relative likelihoods of atleast some of the second codewords having different value of the databit, and to decode the received symbol using the LLR of each data bit.